Representation Of Eigenfunctions Research Articles

Lagrangian multiform structure of discrete and semi-discrete KP systems

A variational structure for the potential AKP system is established using the novel formalism of a Lagrangian multiforms. The structure comprises not only the fully discrete equation on the 3D lattice, but also its semi-discrete variants including several differential-difference equations asssociated with, and compatible with, the partial difference equation. To this end, an overview is given of the various (discrete and semi-discrete) variants of the KP system, and their associated Lax representations, including a novel `generating PDE' for the KP hierarchy. The exterior derivative of the Lagrangian 3-form for the lattice potential KP equation is shown to exhibit a double-zero structure, which implies the corresponding generalised Euler-Lagrange equations. Alongside the 3-form structures, we develop a variational formulation of the corresponding Lax systems via the square eigenfunction representation arising from the relevant direct linearization scheme.

On eigenfunctions of the essential spectrum of the model problem for the Schrödinger operator with singular potential

We are concerned with generalized eigenfunctions of the continuous (essential) spectrum for the Schrödinger operator with singular $\delta$-potential that has support on the sides of an angle in the plane. Operators of this kind appear in quantum-mechanical models for quantum state destruction of two point-interacting quantum particles of which one is reflected by a potential barrier. We propose an approach capable of constructing integral representations for eigenfunctions in terms of the solution of a functional-difference equation with spectral parameter. Solutions of this equation are studied by reduction to an integral equation, with the subsequent study of the spectral properties of the corresponding integral operator. We also construct an asymptotic formula for the eigenfunction at large distances. For this formula a physical interpretation from the point of view of wave scattering is given. Our approach can be used to deal with eigenfunctions in a broad class of related problems for the Schrödinger operator with singular potential. Bibliography: 17 titles.

Tangentially driven active polar linear polymers-An analytical study.

The conformational and dynamical properties of isolated flexible active polar linear polymers (APLPs) are studied analytically. The APLPs are modeled as Gaussian bead-spring linear chains augmented by tangential active forces, both in a discrete and continuous representation. The polar forces lead to linear non-Hermitian equations of motion, which are solved by an eigenfunction expansion in terms of a biorthogonal basis set. Our calculations show that the polymer conformations are independent of activity. However, tangential propulsion strongly impacts the polymer dynamics and yields an active ballistic regime as well as an activity-enhanced long-time diffusive regime, which are both absent in passive systems. The polar forces imply a coupling of modes in the eigenfunction representation, in particular with the translational mode, with a respective strong influence on the polymer dynamics. The total polymer mean-square displacement on scales smaller than the radius of gyration is determined by the active internal dynamics rather than the collective center-of-mass motion, in contrast to active Brownian polymers at large Péclet numbers, reflecting the distinct difference in the propulsion mechanism.

Nonzero-Sum Risk-Sensitive Continuous-Time Stochastic Games with Ergodic Costs

We study nonzero-sum stochastic games for continuous time Markov decision processes on a denumerable state space with risk-sensitive ergodic cost criterion. Transition rates and cost rates are allowed to be unbounded. Under a Lyapunov type stability assumption, we show that the corresponding system of coupled HJB equations admits a solution which leads to the existence of a Nash equilibrium in stationary strategies. We establish this using an approach involving principal eigenvalues associated with the HJB equations. Furthermore, exploiting appropriate stochastic representation of principal eigenfunctions, we completely characterize Nash equilibria in the space of stationary Markov strategies.

Some Spectral Properties of Multiplicative Hermite Equation

We reconstruct the Multiplicative Hermite Equation from multiplicative Sturm-Liouville equation. A new representation of eigenfunctions for the constructed problem are obtained by the power series solution technique. While making these solutions, multiplicative Hermite polynomials were used strongly. We get a generator for multiplicative Hermite polynomials and construct integration representations for these polynomials. Finally, some spectral properties of the multiplicative Hermite problem are examined in detail.

Bilinear equations in Darboux transformations by Boson–Fermion correspondence

Bilinear equation is an important property for integrable nonlinear evolution equation. Many famous research objects in mathematical physics, such as Gromov–Witten invariants, can be described in terms of bilinear equations to show their connections with the integrable systems. Here in this paper, we mainly discuss the bilinear equations of the transformed tau functions under the successive applications of the Darboux transformations for the KP hierarchy, the modified KP hierarchy (Kupershmidt–Kiso version) and the BKP hierarchy, by the method of the Boson–Fermion correspondence. The Darboux transformations are considered in the Fermionic picture, by multiplying the different Fermionic fields on the tau functions. Here the Fermionic fields are corresponding to the (adjoint) eigenfunctions, whose changes under the Darboux transformations are showed to be the ones of the squared eigenfunction potentials in the Bosonic picture, used in the spectral representations of the (adjoint) eigenfunctions. Then the successive applications of the Darboux transformations are given in the Fermionic picture. Based upon this, some new bilinear equations in the Darboux chain are derived, besides the ones of (l−l′) -th modified KP hierarchy. The corresponding examples of these new bilinear equations are given.

An eigenfunction representation of deep waveguides with application to unconventional reservoirs

Guided waves that propagate in deep low-velocity zones can be described using the displacement-stress eigenfunction theory. For a layered subsurface, these eigenfunctions provide a framework to calculate guided-wave properties at a fraction of the time required for fully numerical approaches for wave-equation modeling, such as the finite-difference approach. Using a 1D velocity model representing the low-velocity Eagle Ford Shale, an unconventional hydrocarbon reservoir, we verify the accuracy of the displacement eigenfunctions by comparing with finite-difference modeling. We use the amplitude portion of the Green’s function for source-receiver eigenfunction pairs as a proxy for expected guided-wave amplitude. These response functions are used to investigate the impact of the velocity contrast, reservoir thickness, and receiver depth on guided-wave amplitudes for discrete frequencies. We find that receivers located within the low-velocity zone record larger guided-wave amplitudes. This property may be used to infer the location of the recording array in relation to the low-velocity reservoir. We also study guided-wave energy distribution between the different layers of the Eagle Ford model and find that most of the high-frequency energy is confined to the low-velocity reservoir. We corroborate this measurement with field microseismic data recorded by distributed acoustic sensing fiber installed outside of the Eagle Ford. The data contain high-frequency body-wave energy, but the guided waves are confined to low frequencies because the recording array is outside the waveguide. We also study the energy distribution between the fundamental and first guided-wave modes as a function of the frequency and source depth and find a nodal point in the first mode for source depths originating in the middle of the low-velocity zone, which we validate with the same field data. The varying modal energy distribution can provide useful constraints for microseismic event depth estimation.

Geometric considerations of a good dictionary for Koopman analysis of dynamical systems: Cardinality, “primary eigenfunction,” and efficient representation

Representation of a dynamical system in terms of simplifying modes is a central premise of reduced order modelling and a primary concern of the increasingly popular DMD (dynamic mode decomposition) empirical interpretation of Koopman operator analysis of complex systems. In the spirit of optimal approximation and reduced order modelling the goal of DMD methods and variants are to describe the dynamical evolution as a linear evolution in an appropriately transformed lower rank space, as best as possible. That Koopman eigenfunctions follow a linear PDE that is solvable by the method of characteristics yields several interesting relationships between geometric and algebraic properties. Corresponding to freedom to arbitrarily define functions on a data surface, for each eigenvalue, there are infinitely many eigenfunctions emanating along characteristics. We focus on contrasting cardinality and equivalence. In particular, we introduce an equivalence class, “primary eigenfunctions,’ consisting of those eigenfunctions with identical sets of level sets, that helps contrast algebraic multiplicity from other geometric aspects. Popularly, Koopman methods and notably dynamic mode decomposition (DMD) and variants, allow data-driven study of how measurable functions evolve along orbits. As far as we know, there has not been an in depth study regarding the underlying geometry as related to an efficient representation. We present a construction that leads to functions on the data surface whose corresponding eigenfunctions are efficient in a least squares sense. We call this construction optimal Koopman eigenfunction DMD, (oKEEDMD), and we highlight with examples.

The Analytic Theory of a Monetary Shock

We propose an analytical method to analyze the propagation of a once-and-for-all shock in a broad class of sticky price models. The method is based on the eigenvalue- eigenfunction representation of the dynamics of the cross-sectional distribution of firms’ desired adjustments. A key novelty is that, under assumptions that are appropriate for low-inflation economies, we can approximate the whole profile of the impulse response for any moment of interest in response to an aggregate shock (any displacement of the invariant distribution). We present several applications and discuss extensions. Institutional subscribers to the NBER working paper series, and residents of developing countries may download this paper without additional charge at www.nber.org.

The Analytic Theory of a Monetary Shock

We propose an analytical method to analyze the propagation of a once-and-for-all shock in a broad class of sticky price models. The method is based on the eigenvalue- eigenfunction representation of the dynamics of the cross-sectional distribution of firms’ desired adjustments. A key novelty is that, under assumptions that are appropriate for low-inflation economies, we can approximate the whole profile of the impulse response for any moment of interest in response to an aggregate shock (any displacement of the invariant distribution). We present several applications and discuss extensions.

Toda lattice with corrections via inverse scattering transform

The perturbation theory based on the inverse scattering transform is extended to discuss the Toda lattice with corrections. The time evolution of the associated scattering data is given by some summation representations for corrections and eigenfunctions. The perturbation correction of the conservation laws is investigated. The adiabatic approximate solution and its correction are considered.

A new approach to asymptotic formulas for eigenfunctions of discontinuous non-selfadjoint Sturm–Liouville operators

In the present paper, boundary value problems for discontinuous non-selfadjoint Sturm–Liouville operators on a finite interval with boundary conditions nonlinearly dependent on the spectral parameter are considered, and a new approach for studying the asymptotic representation of the eigenfunctions and their partial derivatives is presented. We obtain the asymptotic representation of the solutions and the eigenvalue, and study some of their main properties. Then, we provide a constructive procedure to obtain the asymptotic form of the eigenfunctions and their partial derivatives in discontinuous case by the canonical form of the Bessel functions $$J_{\frac{1}{2}}(z)$$ , $$J_{\frac{3}{2}}(z)$$ and their derivatives.

Asymptotic Eigenfunctions of the “Bouncing Ball” Type for the Two-Dimensional Schrödinger Operator with a Symmetric Potential

We construct asymptotic eigenfunctions for the two-dimensional Schrodinger operator with a potential in the form of a well that is mirror-symmetric with respect to a line. These functions correspond to librations on this line between two focal points. According to the Maslov complex germ theory, the asymptotic eigenfunctions in the direction transverse to the line with respect to which the well is symmetric have the form of the appropriate Hermite-Gauss mode. We obtain a global Airy-function representation for the asymptotic eigenfunctions in the longitudinal direction.

Modern finite-size criticality: Dirichlet and Neumann boundary conditions

Finite-size critical systems defined on a parallel-plate geometry of finite extent along one single (z) direction with Dirichlet and Neumann boundary conditions at z = 0, L are analyzed in momentum space. We introduce a modified representation for the discrete eigenfunctions in a renormalized one-particle-irreducible (1PI) vertex part scalar field-theoretic framework using either massless or massive fields. The appearance of multiplicities in the Feynman rules to construct diagrams due to this choice of representation of the basis functions is discussed along with the modified normalization conditions. For nonvanishing external quasi-momenta, Dirichlet and Neumann boundary conditions are shown to be unified within a single formalism. We examine the dimensional crossover regimes for these and show a correspondence with those from antiperiodic and periodic boundary conditions. It is demonstrated that finite-size effects for Dirichlet and Neumann boundary conditions do not require surface fields necessarily but are implemented nontrivially from the Feynman rules involving only bulk terms in the Lagrangian. As an application, the critical exponents $\eta$ and $\nu$ are evaluated at least up to two-loop level through diagrammatic means. We show that the critical indices are the same as those from the bulk (infinite) system irrespective of the boundary conditions.

Deep learning for universal linear embeddings of nonlinear dynamics

Identifying coordinate transformations that make strongly nonlinear dynamics approximately linear has the potential to enable nonlinear prediction, estimation, and control using linear theory. The Koopman operator is a leading data-driven embedding, and its eigenfunctions provide intrinsic coordinates that globally linearize the dynamics. However, identifying and representing these eigenfunctions has proven challenging. This work leverages deep learning to discover representations of Koopman eigenfunctions from data. Our network is parsimonious and interpretable by construction, embedding the dynamics on a low-dimensional manifold. We identify nonlinear coordinates on which the dynamics are globally linear using a modified auto-encoder. We also generalize Koopman representations to include a ubiquitous class of systems with continuous spectra. Our framework parametrizes the continuous frequency using an auxiliary network, enabling a compact and efficient embedding, while connecting our models to decades of asymptotics. Thus, we benefit from the power of deep learning, while retaining the physical interpretability of Koopman embeddings.

Boundary Representations of \u03bb-Harmonic and Polyharmonic Functions on Trees

On a countable tree T, allowing vertices with infinite degree, we consider an arbitrary stochastic irreducible nearest neighbour transition operator P. We provide a boundary integral representation for general eigenfunctions of P with eigenvalue λ ∈ C. This is possible whenever λ is in the resolvent set of P as a self-adjoint operator on a suitable ℓ2-space and the diagonal elements of the resolvent (“Green function”) do not vanish at λ. We show that when P is invariant under a transitive (not necessarily fixed-point-free) group action, the latter condition holds for all λ≠ 0 in the resolvent set. These results extend and complete previous results by Cartier, by Figà-Talamanca and Steger, and by Woess. For those eigenvalues, we also provide an integral representation of λ-polyharmonic functions of any order n, that is, functions f: T to mathbb {C} for which (λ ⋅ I − P)nf = 0. This is a far-reaching extension of work of Cohen et al., who provided such a representation for the simple random walk on a homogeneous tree and eigenvalue λ = 1. Finally, we explain the (much simpler) analogous results for “forward only” transition operators, sometimes also called martingales on trees.

Radiation and Exciting Forces of Axisymmetric Structures with a Moonpool in Waves

A highly efficient “hybrid integral-equation method” for computing hydrodynamic added-mass, wave-damping, and wave-exciting force of general body geometries with a vertical axis of symmetry is presented. The hybrid method utilizes a numerical inner domain and a semi-infinite analytical outer domain separated by a vertical cylindrical matching boundary. Eigenfunction representation of velocity potential is used in the outer domain; the three-dimensional potential in the inner domain is solved using a “two-dimensional” boundary element method with ring sources and ring dipoles to exploit the body symmetry for efficiency. With proper solution matching at the common boundary, both radiation and diffraction potentials can be solved efficiently while satisfying the far-field radiation condition exactly. This method is applied to compute the hydrodynamic properties of two different body geometries: a vertical-walled moonpool with a bottom plate that restricts the opening and a spar-like structure with a diverging bottom opening inspired by designs of floating Oscillating Water Columns. The effects of the size of the bottom opening on the hydrodynamic properties of the body are investigated for both geometries. The heave motion of the floater as well as the motion of the internal free surface under incident wave excitation are computed and studied for the spar-like structure.

Characteristics of eigen-system for flexural gravity wave problems

For a class of boundary value problems associated with flexural gravity waves, characteristics of the eigen-system are analysed which are used to study the convergence of the expansion formulae in both cases of single and two layer fluids in water of finite and infinite depths. Using Green’s function technique, the spectral representations of the vertical eigenfunctions are obtained which are used along with the orthogonal mode-coupling relations to prove the convergence of the expansion formulae. Using the expansion formulae flexural gravity wave scattering due to multiple articulations in the presence of compression are investigated. The problem is studied in both the cases of single-layer and two-layer fluids in finite water depth. Effects of compressive force, stiffness of the connectors, length of the elastic plates and water depth, position of interface on wave scattering by articulated plates are studied by analysing the reflection and transmission coefficients. This procedure of proving the convergence of expansion formulae is independent of water depth. The concept and methodology for dealing with wave-structure interaction problems discussed here can be generalised to deal with problems of similar nature arising in the broad area of mathematical physics and engineering.

Enhanced convergence of eigenfunction expansions in convection-diffusion with multiscale space variable coefficients

ABSTRACTA convergence enhancement technique known as the integral balance approach is employed in combination with the Generalized Integral Transform Technique (GITT) for solving diffusion or convection-diffusion problems in physical domains with subregions of markedly different materials properties and/or spatial scales. GITT is employed in the solution of the differential eigenvalue problem with space variable coefficients, by adopting simpler auxiliary eigenproblems for the eigenfunction representation. The examples provided deal with heat conduction in heterogeneous media and forced convection in a microchannel embedded in a substrate. The convergence characteristics of the proposed novel solution are critically compared against the conventional approach through integral transforms without the integral balance enhancement, with the aid of fully converged results from the available exact solutions.

Eigenstates of the quantum Penning-Ioffe nanotrap at resonance

We discuss the choice of physical parameters of a quantum Penning nanotrap under the action of a perturbing inhomogeneous Ioffe magnetic field and also the role of frequency resonance modes. We present a general scheme for constructing the asymptotic behavior of the eigenstates by the generalized geometric quantization method and obtain the reproducing measure in the integral representation of eigenfunctions.